Sorry for the slow reply, this week has been a bit full-on! That's very, very hard for mum.
In the short term, the most you can do may be to offer to help with practical chores that aren't baby-friendly, so that mum can relax or sleep when baby sleeps, not rush off to cut the grass or clean the loo. Food is also a good idea. I know when my husband was away, I ate a lot of something-on-toast and jacket potato type meals, so it was a lovely little boost when a friend made a 'proper meal'. For variety, if nothing else!
For my daughter, the key to getting her to warm to you was to interact with her regularly, but very much at her pace. Smiling and making faces from a safe distance. Then playing with her on the floor, but not touching her. Then touching her but not picking her up. Then picking her up, but not taking her away from mummy. We had a friend who was brilliant at this, and she warmed to him much faster than those who kept trying to push it.
Do you know where baby is sleeping? As I said, my daughter had to be touching me to stay asleep for more than 5-10 minutes, or she woke up really upset that she had lost mummy. I tried holding her hand first (she was in a bedside cot), which worked until I moved my hand in my sleep. A health visitor eventually suggested I seriously consider co-sleeping (following all the safety advice for doing so). It wasn't my first choice for where I wanted baby to sleep, but baby did immediately sleep longer stretches – and so did I once I stopped being quite so nervous about it. That first five hour block of sleep felt amazing.
Another idea might be for mum to record herself singing a lullaby or something and play it on loop for baby when she sleeps, to try and keep her feeling safe and reassured that mum is nearby.